C H A P T E R 15 The biomial distributio Objectives To defie Beroulli sequeces. To review the basic cocepts of the biomial probability distributio. To ivestigate the graph of the biomial probability distributio, ad the effect o the graph of variatio i the values of the parameters. To calculate ad iterpret the mea, variace ad stadard deviatio for the biomial probability distributio. To apply this kowledge of the biomial probability distributio i the solutio of probability-related problems. 15.1 Beroulli sequeces ad the biomial probability distributio A eperimet ofte cosists of repeated trials, each of which may be cosidered as havig oly two possible outcomes. For eample, whe a coi is tossed the two possible outcomes are head ad tail. Whe a die is rolled, the two possible outcomes are determied by the radom variable of iterest for the eperimet. If the evet of iterest is a si, the the two outcomes are si ad ot a si. A Beroulli sequece is the ame used to describe a sequece of trials that possess the followig properties: Each trial results i oe of two outcomes, which are usually desigated either a success, S, or a failure, F. The probability of success o a sigle trial, p, iscostat for all trials, ad thus the probability of failure o a sigle trial is (1 p). The trials are idepedet (so that the outcome o ay trial is ot affected by the outcome of ay previous trial). Cosider, for eample, a player shootig for a goal i etball. The player ca score the goal or miss the goal. If the probability of the player scorig a goal is the same for each attempt at goal, the the player s shots at goal form a Beroulli sequece. 542
Chapter 15 The biomial distributio 543 Eample 1 Suppose that a etball player has a probability of 1 of scorig a goal each time. What is the 2 probability that she will score oe goal from her first two attempts? Solutio If the player scores oe goal from her first two attempts, the the sequece of evets could be either a goal first (G) followed by a miss (M)oramiss first (M) followed by a goal (G). Thus: Pr(oe goal from first two attempts) = Pr(GM) + Pr(MG) = 1 2 1 2 + 1 2 1 2 = 1 2 The biomial probability distributio The umber of successes i a Beroulli sequece of trials is called a biomial radom variable ad is said to have a biomial probability distributio. Eample 2 I a certai coutry the probability of a female child beig bor is 0.52. What is the probability that a family with four childre will have two daughters? Solutio Oe way for a family with four childre to have two daughters is for the two females to be bor first, followed by two males: FFMM where F represets female ad M represets male. The probability of this particular outcome is: 0.52 0.52 0.48 0.48 = (0.52) 2 (0.48) 2 Of course, this is oly oe of the possible ways i which the family could be arraged. How may differet arragemets of FFMM are there? There are 4! arragemets of 4 distict objects i a row. I this case there are repetitios, asthere are two F s ad two M s.toaccout for this divide by 2! for the F s, ad 2! for the M s, givig 4! 2!2!
544 Essetial Mathematical Methods 3&4CAS which is the epasio of 4 4 C 2, usually deoted i the biomial distributio as 2 Thus, the( probability ) of obtaiig eactly two females is give by: 4 (0.52) 2 (0.48) 2 2 This logic ca be repeated to fid the probability of the radom variable X takig ay value from 0 to where there are trials. I geeral, the probability of achievig successes i idepedet trials of a biomial eperimet is Pr(X = ) = p (1 p) = 0, 1,...,! where =!( )! The costats that determie the specific form of a probability distributio are called parameters of the distributio. If a radom variable X has a probability fuctio of this form, the X has a biomial distributio with parameters ad p. Eample 3 Show that the sum of the biomial probabilities is equal to 1. Solutio The sum of the biomial probabilities is p (1 p). =0 The biomial theorem is discussed i Appedi A. This states: ( + a) = i a i i=0 i = + 1 a + 2 a 2 +... 0 1 2 Now, usig the biomial theorem, the sum of the biomial probabilities is give by: p (1 p) = [p + (1 p)] = 0 = (1) = 1 a. See Appedi A for a revisio of the work o permutatios ad combiatios itroduced i Essetial Mathematical Methods 1&2CAS.
Chapter 15 The biomial distributio 545 Eample 4 Fid the probability of obtaiig eactly three heads whe a fair coi is tossed seve times, correct to four decimal places. Solutio Obtaiig a head is cosidered a success here, ad the probability of success o each of the seve idepedet trials is 0.5. Let X be the umber of heads obtaied. I this case the parameters are p = 0.5 ad = 7. 7 Pr(X = 3) = (0.5) 3 (1 0.5) 7 3 = 0, 1,...,7 3 Eample 5 = 35 (0.5) 7 = 0.2734 The probability that a perso curretly i priso has ever bee imprisoed before is 0.72. Fid the probability that of five prisoers chose at radom at least three have bee imprisoed before, correct to four decimal places. Solutio If X is the umber of prisoers who have bee imprisoed before, the 5 Pr(X = ) = (0.72) (0.28) 5 = 0, 1,...,5 ad Pr(X 3) = Pr(X = 3) + Pr(X = 4) + Pr(X = 5) 5 = (0.72) 3 (0.28) 2 5 + (0.72) 4 (0.28) 1 5 + (0.72) 5 (0.28) 0 3 4 5 = 0.8624
546 Essetial Mathematical Methods 3&4CAS Usig the TI-Nspire Use the Biomial Cdf commad from the Probability meu (b 55E) ad complete as show. Use the tab key (e)tomove betwee cells. The result is as show. Usig the Casio ClassPad I tap i the etry lie ad select Calc > Distributio. I the field below Type, tap the ad scroll to select Biomial PD, the tap. Complete the etry scree as show the tap. The calculator returs the aswer as show: Pr(X = 3) = 0.293. To calculate Pr(X 3) we use the fact that Pr(X 3) = 1 Pr(X 2). Tap, chage the distributio to Biomial CD ad complete the etries as show above. The calculator aswer is show. The required aswer is Pr (X 3) = 1 0.1376478 = 0.8623522.
Chapter 15 The biomial distributio 547 Eercise 15A 1 Which of the followig describe a Beroulli sequece? a Tossig a fair coi may times b Drawig balls from a ur cotaiig five red ad three black balls, replacig the chose ball each time c Selectig people at radom from the populatio ad otig their age d Selectig people at radom from the populatio ad otig their se, male or female 2 Evaluate: 6 a (0.3) 4 (0.7) 2 b 4 5 (0.2) 3 (0.8) 2 c 3 5 (0.9) 2 (0.1) 3 2 3 Afair die is rolled si times ad the umber of twos oted. Fid the probability of obtaiig: a eactly 3 twos b more tha 3 twos c at least 3 twos 4 Afair die is rolled five times ad the umber of sies that occur i the five rolls is oted. Fid the probability of: a the first roll beig at si ad the rest ot b eactly oe of the five rolls resultig i a si 5 Afair die is rolled 50 times. Fid the probability of observig: a eactly 10 sies b o more tha 10 sies c at least 10 sies 6 Fid the probability of gettig at least ie successes i 100 trials for which the probability of success is p = 0.1. 7 Afair coi is tossed 50 times. If X is the umber of heads observed, fid: a Pr(X = 25) b Pr(X 25) c Pr(X 10) d Pr(X 40) 8 A supermarket has four checkouts i operatio. A customer is i a hurry ad leaves without makig a purchase if all the checkouts are busy. At that time of day the probability of each checkout beig free is 0.25. Assume that whether or ot a checkout is busy is idepedet of ay other checkout. What is the probability that the customer will make a purchase? 9 A survey of the populatio i a particular city foud that 40% of people regularly participate i sport. What is the probability of fewer tha half of a radom sample of si people regularly participatig i sport? 10 It is kow that the probability of a particular drug causig side effects i a perso is 0.2. What is the probability that at least two of a radom sample of 10 people will eperiece side effects? 11 It is kow that the proportio of voters i a electorate who favour a certai cadidate is 50%. What is the probability that more tha 55% of a sample of 100 voters will favour her?
548 Essetial Mathematical Methods 3&4CAS 12 The maager of a shop kows from eperiece that 60% of her customers will use a credit card to pay for their purchases. Fid the probability that: a the et three customers will use a credit card, ad the three after that will ot b three of the et si customers will use a credit card c at least three of the et si customers will use a credit card d eactly three of the et si customers will use a credit card, give that at least three of the et si customers use a credit card 13 A eamiatio cosists of si multiple-choice questios. Each questio has four possible aswers. At least three correct aswers are required to pass the eamiatio. Suppose a studet guesses the aswer to each questio. a What is the probability the studet guesses every questio correctly? b What is the probability the studet will pass the eamiatio? 14 Records show that, o average, si of the 30 days i November will be raiy. Assumig a biomial distributio, with each day i November as a idepedet trial, fid the probability that et November will have, at most, 10 raiy days. 15 Of the customers who deal with a car retal compay, 45% prefer a automatic car. If there are 35 automatic cars available, what is the probability that the compay will ot be able to meet the demad for automatic cars i a radom group of 80 customers? 16 A multiple-choice test has eight questios, each with five possible aswers, oly oe of which is correct. Fid the probability that a studet who guesses the aswer to every questio will have: a o correct aswers b si or more correct aswers c every questio correct, give they have si or more correct aswers 17 The probability that a full forward i Australia Rules football will kick a goal from outside the 50-metre lie is 0.15. If the full forward has 10 kicks at goal from outside the 50-metre lie, fid the probability that he will: a kick a goal every time b kick at least oe goal c kick more tha oe goal, give that he kicked at least oe goal 18 A sales represetative for a tyre maufacturer claims that the compay s steel-belted radial tyres last at least 55 000 kilometres. A tyre dealer decides to check this claim by testig eight of the tyres. If 75% or more of the eight tyres he tests last at least 55 000 kilometres, he will purchase tyres from the sales represetative. If, i fact, 90% of the steel-belted radials produced by the maufacturer last at least 55 000 kilometres, what is the probability that the tyre dealer will purchase tyres from the sales represetative? 19 A eamiatio cosists of 25 multiple-choice questios. Each questio has four possible aswers. At least 13 correct aswers are required to pass the eamiatio. Suppose the studet guesses the aswer to each questio.
Chapter 15 The biomial distributio 549 a What is the probability that the studet guesses eactly 13 questio correctly? b What is the probability the studet will pass the eamiatio? 20 A multiple-choice test has 20 questios, each with five possible aswers, oly oe of which is correct. Fid the probability that a studet who guesses the aswer to every questio will have: a o correct aswers b 10 or more correct aswers c at least 12 questios correct, give they have 10 or more correct aswers 15.2 The graph of the biomial probability distributio As discussed i Chapter 14, a probability distributio fuctio may be represeted as a formula, a table or a graph. Naturally, each of these will vary as the values of the parameters ad p vary.ithis sectio the behaviour of the graph of the biomial probability distributio will be ivestigated. Cosider for eample the graph of the biomial probability distributio for 20 trials ( = 20) for differig values of the probability of success p. p() 0.2 0.1 p() 0.2 p = 0.2 0 2 4 6 8 10 12 14 16 18 20 p = 0.5 0.1 0 2 4 6 8 10 12 14 16 18 20
550 Essetial Mathematical Methods 3&4CAS p() 0.2 p = 0.8 0.1 0 2 4 6 8 10 12 14 16 18 20 By comparig the three graphs it ca be see that: Whe p = 0.2 the graph is positively skewed, idicatig that mostly from 1 to 8 successes are observed i the 20 trials. Whe p = 0.5 the graph is symmetrical (which is as epected whe the probability of success is the same as the probability of failure), idicatig that mostly from 6 to 14 successes are observed i the 20 trials. Whe p = 0.8 the graph is egatively skewed, idicatig that mostly from 12 to 19 successes are observed i the 20 trials. A method for plottig a biomial distributio with a CAS calculator ca be foud i Appedi B. Eercise 15B 1 From the graphs o page 518 ad above, fid the most likely outcome from a biomial eperimet whe: a = 20 ad p = 0.2 b = 20 ad p = 0.5 c = 20 ad p = 0.8 2 The followig table gives the biomial distributio for = 10 ad various values of p give to two decimal places: 0 1 2 3 4 5 6 7 8 9 10 p = 0.3 0.03 0.12 0.23 0.27 0.20 0.10 0.04 0.01 0.00 0.00 0.00 p = 0.6 0.00 0.00 0.01 0.04 0.11 0.20 0.25 0.21 0.12 0.04 0.01 p = 0.9 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.06 0.19 0.39 0.35 Sketch the graph of the biomial distributio for each of the values of p o the same aes, ad write a descriptio of the plots obtaied. 3 The followig table gives the biomial distributio for = 6 ad p = 0.5 ad for = 10 ad p = 0.5. (Note that because of roudig error the probabilities do ot sum eactly to 1.)
Chapter 15 The biomial distributio 551 0 1 2 3 4 5 6 7 8 9 10 = 6 p = 0.5 0.02 0.09 0.23 0.31 0.23 0.09 0.02 0.00 0.00 0.00 0.00 = 10 p = 0.5 0.0 0.01 0.04 0.12 0.21 0.25 0.21 0.12 0.04 0.01 0.00 Sketch the graph of the biomial distributio for each set of values of ad p o the same aes, ad write a descriptio of the plots obtaied. 4 Plot the probability distributio fuctio Pr(X = ) = p (1 p) for = 8 ad p = 0.25. 5 Plot the probability distributio fuctio Pr(X = ) = p (1 p) for = 12 ad p = 0.35. = 0, 1,..., = 0, 1,..., 6 a Plot the probability distributio fuctio Pr(X = ) = p (1 p) = 0, 1,..., b c for = 10 ad p = 0.2. O the same aes plot Pr(X = ) = p (1 p) = 0, 1,..., for = 10 ad p = 0.8, usig a differet plottig symbol. Compare the two distributios. 7 a Plot the probability distributio fuctio Pr(X = ) = p (1 p) = 0, 1,..., b for = 20 ad p = 0.2. O the same aes plot Pr(X = ) = p (1 p) = 0, 1,..., for = 20 ad p = 0.5, usig a differet plottig symbol.
552 Essetial Mathematical Methods 3&4CAS c d O the same aes plot Pr(X = ) = p (1 p) = 0, 1,..., for = 20 ad p = 0.8, usig a differet plottig symbol. Commet o the effect of the value of p o the shape of the distributio. 15.3 Epectatio ad variace How may heads would you epect to obtai, o average, if a fair coi was tossed 10 times? While the eact umber of heads i the 10 tosses would vary, ad could theoretically take values from 0 to 10, it seems reasoable that the log-ru average umber of heads would be 5. It turs out that this is correct. That is, for a biomial radom variable whe = 10 ad p = 0.5: E(X) = p() = 5 I geeral, the epected value for a biomial radom variable is equal to the umber of trials multiplied by the probability of success. The variace may also be calculated from the parameters ad p. I geeral, if X is the umber of successes i trials, each with probability of success p, the the epected value ad variace of X are: E(X) = p Var(X) = p(1 p) While it is ot ecessary i this course to be familiar with the derivatios of these formulae, they are icluded for completeess. First of all, cosider the epected value of X. Bydefiitio: E(X) = p() = p (1 p) (substitutig the distributio formula) =0 )! = p (1 p) (epadig! ( )! =0! = p (1 p) (sice the first term will equal! ( )! =1 zero whe = 0) ( )! = p (1 p) (sice! = ( 1)!) ( 1)! ( )! =1 ( )! = p (1 p) (cacellig the s) ( 1)! ( )! =1
Chapter 15 The biomial distributio 553 This epressio is very similar to the probability fuctio for a biomial radom variable, ad the sum of a probability fuctio p() isequal to 1. Takig factors of ad p from the epressio ad lettig z = 1gives: 1 1 E(X) = p p 1 (1 p) =1 1 1 1 = p p z (1 p) 1 z z=1 z 1 1 Note that p(z) = p z (1 p) 1 z is the sum of all values of the probability z=1 z fuctio for a biomial radom variable z, which is the umber of successes i 1 trials each with probability of success p, ad is equal to 1. Thus, for a biomial radom variable: E(X) = p The variace of the biomial radom variable may be foud usig: Var(X) = 2 = E(X 2 ) 2, where = p Thus to fid 2, first determie E(X 2 ). E(X 2 ) = 2 p (1 p) =0 = 2! p (1 p)!( )! =0 Note that this time 2 does ot appear as a factor of!. The strategy used here is to first determie E[X(X 1)]. E[X(X 1)] = ( 1) p (1 p) =0! = ( 1) p (1 p)! ( )! =0! = ( 1) p (1 p)! ( )! =2 sice the first ad secod terms of this sum equal zero (whe = 0 ad = 1). Agai, this epressio is very similar to the biomial distributio formula, but is ot quite equal to the sum of a probability fuctio p(). Takig out factors of ( 1)p 2 ad lettig z = 2gives ( ) E[X(X 1)] = ( 1)p 2 ( 2)! p 2 (1 p) ( 2)! ( )! = 2( ) 2 = ( 1)p 2 2 p z (1 p) 2 z z z=0
554 Essetial Mathematical Methods 3&4CAS Agai, p(z) = 2 2 p z (1 p) 2 z is the sum of all values of the probability z = 0 z fuctio for biomial radom variable z, which is the umber of successes i 2 trials each with probability of success p ad is thus equal to 1, ad E[X(X 1)] = ( 1)p 2 ad E[X(X 1)] = E(X 2 ) E(X) ( 1)p 2 = E(X 2 ) p E(X 2 ) = ( 1)p 2 + p which is a epressio for E(X 2 )iterms of ad p as required. Thus: Eample 6 Var( X) = E(X 2 ) 2 = ( 1)p 2 + p (p) 2 = p(1 p) A eamiatio cosists of 30 multiple-choice questios, each questio havig three possible aswers. If a studet guesses the aswer to every questio, how may will she epect to get right? Solutio Sice the umber of correct aswers is a biomial radom variable, with parameters p = 1 ad = 30, the studet who guesses has a epected result of = p = 10 3 correct aswers (ot eough to pass if the pass mark is 50%!). Eample 7 The probability of cotractig iflueza this witer is kow to be 0.2. Of the 100 employees at a certai busiess how may would the ower epect to get iflueza? Fid the stadard deviatio of the umber who will get iflueza ad calculate ± 2. Iterpret the iterval [ 2, + 2 ] for this eample. Solutio The umber of people who get iflueza is a biomial radom variable, with parameters p = 0.2 ad = 100. The ower will epect = p = 20 of her employees to cotract iflueza, with a variace ad hece a stadard deviatio = 16 = 4 2 = p(1 p) = 100 0.2 0.8 = 16
Chapter 15 The biomial distributio 555 The ± 2 = 20 ± (2 4) = 20 ± 8orfrom 12 to 28 Thus the ower of the busiess kows there is a probability of about 0.95 that from 12 to 28 of her employees will cotract iflueza this witer. Eercise 15C 1 Fid the mea ad variace of each of the biomial radom variables with parameters: a = 25 ad p = 0.2 b = 10 ad p = 0.6 c = 500 ad p = 1 3 d = 40 ad p = 20% 2 Whe a fair die is rolled si times, fid: a the epected value for the umber of sies obtaied b the probability that more tha the epected umber of sies is obtaied 3 The survival rate for a certai disease is 75%. Of the et 50 people who cotract the disease, how may would you epect would survive? 4 A biomial radom variable has a mea 12 ad variace 9. Fid the parameters ad p, ad hece Pr(X = 7). 5 A biomial radom variable has a mea 30 ad variace 21. Fid the parameters ad p, ad hece Pr(X = 20). 6 Afair coi is tossed 20 times. Fid the mea ad stadard deviatio of the umber of heads obtaied ad calculate ± 2. Iterpret the iterval [ 2, + 2 ] for this eample. 7 Records show that 60% of the studets i a certai state atted govermet schools. If a group of 200 studets are to be selected at radom, fid the mea ad stadard deviatio of the umber of studets i the group who atted govermet schools, ad calculate ± 2. Iterpret the iterval [ 2, + 2 ] for this sample. 15.4 Usig the CAS calculator to fid the sample size Eample 8 The probability of wiig a prize i a game of chace is 0.45. What is the least umber of games that must be played to esure that the probability of wiig at least twice is more tha 0.95?
556 Essetial Mathematical Methods 3&4CAS Solutio Sice the probability of wiig each game is the same each time the game is played, this is a eample of a biomial distributio, with the probability of success p = 0.45. The required aswer is value of such that: Pr(X 2) > 0.95 Equivaletly, 1 Pr(X < 2) > 0.95 Pr(X < 2) < 0.05 Pr(X < 2) = Pr(X = 0) + Pr(X = ( 1) ) = 0.45 0 (0.55) + 0.45 1 (0.55) 1 0 1 = (0.55) + 0.45 1 (0.55) 1 sice = 1 ad = 0 1 The aswer is the value of such that: (0.55) + 0.45 1 (0.55) 1 < 0.05 This is ot a equatio that ca be solved algebraically, but a CAS calculator ca be used to solve this equatio. The calculator solves the equatio (0.55) + 0.45(0.55) 1 = 0.05 Numerically = 8.4754... Thus the game must be played at least ie times to esure that the probability of wiig at least twice is more tha 0.95. Eercise 15D 1 The probability of a target shooter hittig the bullseye o ay oe shot is 0.2. a If the shooter takes five shots at the target, fid the probability of: i missig the bullseye every time ii hittig the bullseye at least oce b What is the smallest umber of shots the shooter should make to esure a probability of more tha 0.95 of hittig the bullseye at least oce? 2 The probability of wiig a prize with a lucky ticket o a wheel of fortue is 0.1. a If a perso buys 10 lucky tickets, fid the probability of: i wiig twice ii wiig at least oe prize b What is the smallest umber of tickets that should be bought to esure a probability of more tha 0.7 of wiig at least oce? 3 Re is shootig at a target. His probability of hittig the target is 0.6. What is the miimum umber of shots eeded for the probability of Re hittig the target eactly five times to be more tha 25%?
Chapter 15 The biomial distributio 557 4 Jaet is selectig chocolates at radom out of a bo. She kows that 20% of the chocolates have hard cetres. What is the miimum umber of chocolates she eeds to select to esure that the probability of choosig eactly three hard cetres is more tha 10%? 5 The probability of wiig a prize i a game of chace is 0.35. What is the fewest umber of games that must be played to esure that the probability of wiig at least twice is more tha 0.9? 6 Geoff has determied that his probability of hittig 4 off ay ball whe playig cricket is 0.07. What is the fewest umber of balls he must face to esure that the probability of hittig more tha oe 4 is more tha 0.8? 7 Moique is practisig goalig for etball. She kows from past eperiece that her chaces of makig ay oe shot is about 70%. Her coach has asked her to keep practisig util she scores 50 goals. How may shots would she eed to attempt to esure that the probability of makig at least 50 shots is more tha 0.99?
558 Essetial Mathematical Methods 3&4CAS Review Chapter summary A Beroulli sequece describes the repetitios of a eperimet where: Each trial results i oe of two outcomes, which are usually desigated either a success, S,orafailure, F. The probability of success o a sigle trial, p, iscostat for all trials (ad thus the probability of failure o a sigle trial is (1 p). The trials are idepedet (so that the outcome o ay trial is ot affected by the outcome of ay previous trial). If X is the umber of successes i Beroulli trials, the X is called a biomial radom variable ad is said to have a biomial probability distributio with parameters ad p. I geeral, the probability of observig successes i idepedet trials of a biomial eperimet is: Pr(X = ) = p (1 p) = 0, 1,...,! where =!( )! If X has a biomial probability distributio with parameters ad p: E(X) = p Var(X) = p(1 p) The shape of the graph of the biomial probability fuctio depeds o the values of ad p. p() 0.2 0.1 p = 0.2 0 2 4 6 8 10 12 14 16 18 20
Chapter 15 The biomial distributio 559 p() 0.2 0.1 p() p = 0.5 0 2 4 6 8 10 12 14 16 18 20 0.2 p = 0.8 0.1 0 2 4 6 8 10 12 14 16 18 20 Multiple-choice questios 1 A coi is biased so that the probability of a head is 0.6. The probability that eactly three heads will be observed whe the coi is tossed five times is: A 0.6 3 B (0.6) 3 C (0.6) 3 (0.4) 2 D 10 (0.6) 3 (0.4) 2 E 5 C 3 (0.6) 5 2 The probability that the 8:25 trai arrives o time is 0.35. What is the probability that the trai is o time at least oce durig a workig week (Moday to Friday)? A 1 (0.65) 5 B (0.35) 5 C 1 (0.35) 5 D 5 (0.35) 1 (0.65) 4 E (0.65) 5 3 Afair die is rolled four times. The probability that a umber greater tha 4 is observed o two occasios is: A 1 B 16 C 1 D 1 E 8 4 81 9 81 27 4 The probability a perso i a certai tow has a tertiary educatio is 0.4. What is the probability that if 80 people are chose at radom from this tow, less tha 30 will have a tertiary educatio? A 0.7139 B 0.2861 C 0.0827 D 0.3687 E 0.3750 Review
560 Essetial Mathematical Methods 3&4CAS Review 5 If X is a biomial radom variable with parameters = 18 ad p = 1, the the mea ad 3 variace of X are closest to: A = 6, 2 = 4 B = 9, 2 = 4 C = 6, 2 = 2 D = 6, 2 = 16 E = 18, 2 = 6 6 Which oe of the followig graphs best represets the shape of a biomial probability distributio of the radom variable X with 10 idepedet trials ad probability of success 0.7? A B C D E 7 Suppose that X is a biomial radom variable with mea = 10 ad stadard deviatio = 2. The the probability of success, p, iay trial is: A 0.4 B 0.5 C 0.6 D 0.7 E 0.8 8 Suppose that X is the umber of heads observed whe a coi kow to be biased towards heads is tossed 10 times. If Var(X) = 1.875, the the probability of a head o ay oe toss is: A 0.25 B 0.55 C 0.75 D 0.65 E 0.80 Questios 9 ad 10 refer to the followig iformatio: The probability of Thomas beatig William i a set of teis is 0.24, ad Thomas ad William decide to play a set of teis every day for days. 9 What is the fewest umber of days o which they should play to esure that the probability of Thomas wiig at least oe set is more tha 0.95? A 7 B 8 C 9 D 10 E 11 10 What is the fewest umber of days o which they should play to esure that the probability of Thomas wiig at least two sets is more tha 0.95? A 12 B 18 C 17 D 21 E 14 Short-aswer questios (techology-free) 1 If X is a biomial radom variable with parameters = 4 ad p = 1 3, fid: a Pr(X = 0) b Pr(X = 1) c Pr(X 1) d Pr(X 1) 2 A salesperso kows that 60% of the people who eter a particular shop will make a purchase. What is the probability that of the et three people who eter the shop eactly two will make a purchase?
Chapter 15 The biomial distributio 561 3 If 10% of patiets fail to improve o a certai medicatio, fid the probability that of five patiets selected at radom oe or more will fail to show improvemet. 4 A machie has a probability of 0.1 of maufacturig a defective part. a What is the epected umber of defective parts i a radom sample of 20 parts maufactured by the machie? b What is the stadard deviatio of the umber of defective parts? 5 A eperimet cosists of four idepedet trials. The probability of success i a trial is p. Each trial eds i either a success or a failure. Fid the probability of each of the followig i terms of p: a o successes b oe success c at least oe success d four successes e at least two successes 6 A coi is tossed 10 times. The probability of three heads is ( 1 2 )10. State the value of. 7 A eperimet cosists of five idepedet trials. Each trial results i a success or failure. The probability of success i a trial is p. Fid i terms of p the probability of eactly oe success give at least oe success. 8 A dice is tossed five times. What is the probability of obtaiig a eve umber o the uppermost face o three of the tosses? 9 I a particular city the probability of rai o ay day i Jue is 1. What is the probability of 5 it raiig o three of five days? Eteded-respose questios 1 I a test to detect learig disabilities, a child is give 10 questios, each of which has possible aswers labelled A, B ad C. Childre with a disability of type 1 almost always aswer A or B o every questio, while childre with a disability of type 2 almost always aswer C o every questio. Childre without either disability have a equal chace of aswerig A, B or C for each questio. a What is the probability that the aswers give by a child without either disability will be all A s ad B s, thereby idicatig a type 1 disability? b A child is further tested for type 2 disability if he or she aswers C five or more times. What is the probability that a child without either disability will test positive for type 2 disability? 2 A ispector takes a radom sample of 10 items from a very large batch. If oe of the items is defective he accepts the batch, otherwise he rejects the batch. What is the probability that a batch is accepted if the fractio of defective items is 0, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1? Plot these probabilities agaist the correspodig fractio defective. Is the ispectio method is a good oe or ot? Review
562 Essetial Mathematical Methods 3&4CAS Review 3 It has bee foud i the past that 4% of the compact disks produced i a certai factory are defective. A sample of 10 is draw radomly from each hour s productio ad the umber of defectives is oted. a I what percetage of these hourly samples would there be at least two defectives? b Fid the mea ad stadard deviatio of the umber of defectives oted, ad calculate ± 2. c A particular sample is foud to cotai three defectives. Would this cause you to have doubts about the productio process? 4 A pizza compay claims that they deliver 90% of orders withi 30 miutes. I a particular 2-hour period the supervisor otes that 67 pizzas are ordered ad 12 are delivered late. If the delivery compay is correct, ad 90% of pizza are delivered o time, what is the probability that at least 12 pizzas are delivered late? 5 a A sample of si objects is to be draw from a large populatio i which 20% of the objects are defective. Fid the probability that the sample cotais: i three defectives ii less tha three defectives b Aother large populatio cotais a proportio p of defective items. i Write dow a epressio of p for P, the probability that a sample of si cotais eactly two defectives. ii By differetiatig to fid dp dp, show that P is greatest whe p = 1 3. 6 Groups of si people are chose at radom ad the umber,,ofpeople i each group who ormally wear glasses is recorded. The results obtaied from 200 groups of si are show i the table: No. i group wearig glasses () 0 1 2 3 4 5 6 No. of occurreces (p) 17 53 65 45 18 2 0 a Calculate, from the above data, the mea value of. b Assumig that the situatio ca be modelled by a biomial distributio havig the same mea as the oe calculated above, state the appropriate values for the biomial parameters ad p. c Calculate the theoretical frequecies correspodig to those i the table. 7 A samplig ispectio scheme is devised as follows. A sample of size 10 is draw at radom from a large batch of articles ad all 10 articles are tested. If the sample cotais less tha two faulty articles the batch is accepted; if the sample cotais three or more faulty articles, the batch is rejected; but if the sample cotais eactly two faulty articles, a secod sample of size 10 is take ad tested. If this secod sample cotais o faulty articles, the batch is accepted; but if it cotais ay faulty articles, the batch is rejected. Previous eperiece has show that 5% of the articles i a batch are faulty.
Chapter 15 The biomial distributio 563 a Fid the probability that the batch is accepted after the first sample is take. b Fid the probability that the batch is rejected. c Fid the epected umber of articles to be tested. 8 It may be assumed that dates of birth i a large populatio are distributed throughout the year so that a probability of a radomly chose perso s date of birth beig i ay particular moth may be take as 1 12. a b c Fid the probability that of si people chose at radom eactly two will have birthday i Jauary. Fid the probability that of eight people chose at least oe will have a birthday i Jauary. N people are chose at radom. Fid the least value of N so that the probability that at least oe will have a birthday i Jauary eceeds 0.9. 9 Suppose that, i flight, aeroplae egies fail with probability q, idepedetly of each other, ad a plae will complete the flight successfully if at least half of its egies are still workig. For what values of q is a two-egie plae to be preferred to a four-egie oe? Review